Strain Energy Method - Civil Engineering (CE) PDF Download

 Deflection by Strain Energy Method 

The concepts of strain, strain-displacement relationships are very useful in computing energy-related quantities such as work and strain energy. These can then be used in the computation of deflections. In the special case, when the structure is linear elastic and the deformations are caused by external forces only, (the complementary energy U* is equal to the strain energy U) the displacement of structure in the direction of force P_j is expressed by

   (4.16)

This equation is known as Castigliano's theorem. It must be remembered that its use is limited to the calculation of displacement in linear elastic structures caused by applied loads. The use of this theorem is equivalent to the virtual work transformation by the unit-load theorem.

Calculation of Strain Energy

When external loads are applied on an elastic body they deform. The work done is transformed into elastic strain energy U that is stored in the body. We will develop expressions for the strain energy for different types of loads.

Axial Force : Consider a member of length L and axial rigidity AE subjected to an axial force P applied gradually as shown in the Figure 4.24. The strain energy stored in the member will be equal to the external work done by the axial force i.e

   (4.17)

Figure 4.24 Member subjected to axial force

Bending Moment: Consider a beam of length L and flexural rigidity EI subjected to a general loading as shown in Figure 4.25. Consider a small differential element of length, dx.The energy stored in the small element is given by

    (4.18)

The total strain energy in the entire beam will be

   (4.19)

Figure 4.25 Member under bending

Shear Force: The strain energy stored in the member due to shearing force is expressed by

       (4.20)

where V is the shearing force; and GA₅ is the shearing rigidity of the member.

Twisting Moment: The strain energy stored in the member due to twisting moment is expressed by

       (4.21)

where T is the twisting moment; and GJ is the torsional rigidity of the member.

Example 4.18 Find the horizontal deflection at joint C of the pin-jointed frame as shown in Figure 4.26(a). AE is constant for all members.

Solution: The force in various members of the frame is shown in Figure 4.26(b). Calculation of strain energy of the frame is shown in Table 4.4.

Table 4.4

Member	Length ( L )	Force ( P )
AB	L	P	P2L/2AE
BC	L	P	P2L/2AE
BD	√2L	-P√2	√2P2L/2AE
CD	L	0	0

∑(√2+1)P²L/AE

Horizontal displacement of joint C, 

Example 4.19 A bar of uniform cross-section is bent into a quadrant of circle of radius R. One end of the bent is fixed and other is free. At the free end it carries a vertical load W. Determine the vertical and horizontal deflection at A.

Solution:

Vertical displacement of A : The vertical displacement of A is given by

For evaluation of the total strain energy in the system, consider a small element Rdθ as shown in the Figure. The bending moment at this element,M_θ = -Rsinθ,Thus

Since there is no horizontal force acting at point A , apply a horizontal force, F at A as shown in Figure 4.27(b). From the Castigliano's theorem, the horizontal displacement of A due to applied external load W is given by 

The bending moment at the small element Rdθ is M_θ= -WRsinθ-FR(1-cosθ). Thus, the horizontal displacement of A

 (i.e. deflection is in direction)

Example 4.20 Determine the deflection of the end A of the beam as shown in Figure 4.28. The flexibility of the spring is f=L³/EI.

Solution: Reactions at support B and C are R_B=1.5P (upward) and R_c=0.5P (downward) Force in the spring = Reaction, R_c=0.5P

Deflection under the load is given by

where U=U_AB + U_BC + U_S is the total strain energy stored in the system; U_AB is the energy stored in the member AB;U_BC is the energy stored in the member BC ; and U_S=strain energy in the spring. 

Strain energy in the spring is given by

Consider member AB : (x measured from A)

M_x = -Px

Consider member BC : (x measured from C)

Thus,

U=U_AB + U_BC + U_S

The deflection of point A,